Lesson 16

The purpose of this warm-up is to begin looking at the different triangles that can be drawn when three measures are specified. The first set of triangles in this activity all share the same 3 side lengths. The second set of triangles all share the same 3 angle measures. Later in this lesson, students will look at sets of triangles that share some combination of side lengths and angle measures.

Provide access to geometry toolkits. Give students 1 minute of quiet think time, followed by a whole-class discussion.

Some students may say that all the triangles in the second set are “the same shape.” This statement can result from two very different misconceptions. Listen to the students’ reasoning and explain as needed:

1. Just because they are all in the same category “triangles” doesn’t mean they are all the same shape. If we can take two shapes and position one exactly on top of the other, so all the sides and corners line up, then they are identical copies.
2. These triangles are scaled copies of each other, but that does not make them “the same” because their side lengths are still different. Only scaled copies made using a scale factor of 1 are identical copies.

Invite students to share things they notice, things that are the same and things that are different about the triangles. Record and display these ideas for all to see.

If these discussion points do not come up in students’ explanations make them explicit:

In the first set:

• All the triangles are identical copies, just in different orientations.
• They have the same 3 side lengths.
• They have the same 3 angle measures (can be checked with tracing paper or a protractor).

In the second set:

• The triangles are not identical copies.
  • Note: Students may recognize that these triangles are scaled copies of each other, since they have the same angle measures. However, this is the first time students have seen scaled copies in different orientations, and it is not essential to this lesson that students recognize that these triangles are scaled copies.
• They have the same 3 angle measures.
• They have different side lengths (can be checked with tracing paper or a ruler).

The goal is to make sure students understand that the second set has 3 different triangles (because they are different sizes) and the first set really only shows 1 triangle in many different orientations. Tracing paper may be helpful to convince students of this.

In this activity, students examine different orientations of triangles that all share 2 sides lengths and one angle measure. They recognize that some of these triangles are identical copies and others are different triangles (not identical copies).

In the coming lessons, students are asked to draw their own triangles. On their own, students often have trouble thinking about triangles where the three given conditions are not included adjacent to one another. For example, when given two sides and an angle, many students will immediately think of putting the given angle between the two sides, but struggle with visualizing putting the angle anywhere else. This task is important for helping students view this as a viable option.

Arrange students in groups of 2. Give students 2–3 minutes of quiet work time followed by time to discuss their explanations with a partner. Follow with a whole-class discussion. Provide access to geometry toolkits.

Some students may say that there are 9 different triangles, because they do not recognize that some of them are identical copies oriented differently. Prompt them to use tracing paper to compare the triangles.

Select students to share the similarities and differences between the triangles in the set.

Trace a few of the triangles from the set and show how you can turn, flip, or move some of them to line up while others cannot be lined up. Ask students what this means about all the triangles in the set (they are not all identical to each other). Explain that, “While there are certainly times when the position of a triangle is important (‘I wouldn’t want my roof upside down!’), for this unit in geometry, we will consider shapes the same if they are identical copies.”

To highlight the differences among the triangles, ask students: 

• “Is there only one possible triangle that could be created from the given conditions?” (No, there were 4.)
• “How would you explain what is different about these four triangles?” (Some have the \(30^\circ\) angle between the two sides of known length and others have the \(30^\circ\) angle next to the side of unknown length.)

Explain to students that it seems the order in which the conditions are included in the triangle (for example, is the angle between the two sides or not?) matters in creating different triangles. Emphasize that the three required pieces (2 sides and 1 angle) do not have to all be put next to one another. When they are asked to draw triangles with three or more conditions, they should consider the way in which the conditions are arranged in their drawing. For example, think about whether the given angle must go between the two sides or not.

This activity is similar to what students did in the previous activity; however, here the conditions given are 2 angles and 1 side.

Keep students in the same groups. Tell students that this activity is similar to the previous one, and they should pay close attention to what they find different here. Provide access to geometry toolkits. Give students 2–3 minutes of quiet work time followed by time to discuss their explanations with a partner. Follow with a whole-class discussion.

Some students may say there are only 2 different triangles in this set, because they do not notice the slight size difference between the smaller two groups of triangles. Prompt them to look at where the \(80^\circ\) angle is located in comparison to the 6 cm side.

Ask a few students to share how many different triangles they think are in this set. Select students to share the similarities and differences between triangles in the set. If necessary, trace a few triangles from the set and show how you can turn, flip, or move some of them to line up while others cannot be lined up. Ask students what this means about all the triangles in the set (they are not all identical to each other).

To highlight the differences among the triangles ask students:

• “What differences do you see between the triangles in this activity and the triangles in the previous activity?” (The given conditions here were 2 angles and 1 side, the previous activity was 2 sides and 1 angle.)
• “What similarities do you see between the triangles in this activity and the triangles in the previous activity?” (These triangles have all the same conditions but in a different order, and they made different triangles as was seen in the previous activity.)

If time permits, consider asking students to use a protractor to measure the unlabeled angle from each of the three different triangles. Discuss what they notice about the third angle. (It’s the same size in every triangle.)

Explain to students that here we see another example of different triangles that can be made using the same conditions (2 angles and 1 side) in different orders (side between the two angles, side next to the 40 degree angle and side next to the 80 degree angle). Tell them that in upcoming lessons we will continue to investigate what they noticed here with the addition of drawing the different triangles.